Reconstruction methods for infrared absorption imaging

Abstract

In infrared absorption imaging, the requirement is to reconstruct the spatial distribution of the optical absorption coefficient, from boundary measurements of the flux intensity of light arising from a specific source distribution. An accurate and efficient model is required to simulate data for given experimental conditions and for any hypothesized solution (the Forward Problem). The Inverse Problem is then to derive the solution that best fits the data, subject to constraints imposed by a priori knowledge (e.g. positivity). The Forward Problem is denoted (chi) equals A(mu) + n where (mu) is the required functional map, (chi) the boundary data, A the Forward Transform and n noise, and the Inverse Problem is (mu) equals A+(chi) , where A+ is an approximation to the Inverse Transform. The experimental arrangement modelled assumes an inhomogeneous cylindrical object. A picosecond dye laser produces input pulses at N locations and a time resolved detector makes measurements at N output sites. This (chi) is an N2 by 1 vector and (mu) can be reconstructed to, at best, an N by N image. The Forward Model described here is an analytic approach using the Greens Function of the diffusion equation in a cylinder, (the P1 approximation to the radiative transfer equation). It may be parameterized by the global values of absorption and scattering coefficients ((mu) a, and (mu) s), which have to be adjusted to best fit the data. The Inverse Problem is highly ill-posed. To solve it, we use the Moore-Penrose generalized inverse A+ equals (A*A)-1A*, and two simple regularization techniques: truncated singular value reconstruction and Tikhovov regularization. Examination of the singular vectors of the kernel demonstrate that the solution is dominated by surface effects, unless a very high signal-to-noise ratio is obtained in the data. Results are shown for simulated mathematical phantoms and a tissue-equivalent phantom composed of polystyrene microspheres.